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Mass Transport Between Stationary Random Measures

Khezeli, Ali | 2016

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  1. Type of Document: Ph.D. Dissertation
  2. Language: Farsi
  3. Document No: 48852 (05)
  4. University: Sharif University of Technology
  5. Department: Pure Mathematics
  6. Advisor(s): Alishahi, Kasra; Haji Mirsadeghi, Mir-Omid
  7. Abstract:
  8. Given two stationary random measures on Rd, we study transport kernels between them that are translation-covariant. We will provide an algorithm that constructs such a transport kernel, with some assumption on their intensities. As a result, this algorithm can be used to construct the Palm version of an ergodic random measure by simply applying a (random) translation and vice versa, to reconstruct the distribution of the random measure from its Palm distribution. Given realizations of the two random measures, our algorithm provides its result in a deterministic way. The existence of such transport kernels is proved in [18] in an abstract way. Nevertheless, there has been tremendous interest in constructive algorithms in recent years. Our method is a generalization of [6] which constructs a transport kernel between the Lebesgue measureand an ergodic point process in Rd. In the general case, we define and limit ourselves to constrained transport kernels and constrained transport densities. We will also define stability of constrained transport densities inspired by the notion of stable matchings in graphs. Limiting to constrained transport densities is a key point in our algorithm and the result of the algorithm is stable. For stable constrained transport densities, we study some properties like existence and uniqueness, onotonicity in terms of the two measures and boundedness. Constructing an allocation (i.e. a translation-covariant transport map) has been of more interest in the literature. We will study conditions for existence of allocations, study special cases that our algorithm yields an allocation and provide other algorithms to construct allocations under some assumptions. As a tool for proving some results, we introduce and study some properties of Voronoi transport kernel for a measure, which is a generalization of Voronoi tessellation
  9. Keywords:
  10. Stationary State ; Mass Transfer ; Capacity Constraint ; Random Measure ; Palm Distribution ; Stable Matching ; Voronoi Transport Kernel ; Shift Coupling

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